Question: In the diagram below, $BC$ is 8 cm.  In square centimeters, what is the area of triangle $ABC$?

[asy]
defaultpen(linewidth(0.7));
draw((0,0)--(16,0)--(23,20)--cycle);
draw((16,0)--(23,0)--(23,20),dashed);
label("8 cm",(8,0),S);
label("10 cm",(23,10),E);
label("$A$",(23,20),N);
label("$B$",(0,0),SW);
label("$C$",(16,0),SE);
[/asy]
The area of a triangle is $\frac{1}{2}(\text{base})(\text{height})$, so the area of triangle $ABC$ is $\frac{1}{2}(8\text{ cm})(\text{10 cm})=\boxed{40}$ square centimeters.